ABSTRACT
The procedures developed in Chapter 2 have provided a collection of important tools for inverting generating functions. Already, there are many generating functions that we can invert using a combination of the scaling and sliding tools, in combination with the addition and convolution principles. These tools include the ability to identify and manipulate the generating functions of many commonly used discrete probability distributions. However, there is an important class of generating functions that we have not attempted to invert, but that occur very frequently in the solution of difference equations. In fact, this class is among the most commonly occurring difference equations that one encounters. In addition, they have historically provided important complications in the solution of difference equations. This class of generating functions are those that contain general polynomials in the denominator, and will be the subject of this entire chapter. At the conclusion of the chapter, the reader should be able to invert a generating function whose denominator is a polynomial in s.
